## Frank Morgan's Math Chat |

**THE MATH CHAT TV** student staff spent time last month with the Williamstown
Elementary School classes of Susan Pleines (fourth grade) and Lynn
Bernstein (fifth grade). Michael Biaocchi's group studied vectors and used
them for treasure hunts. Rita Forte's group compiled basketball statistics
and created a new game. Fumi Tosu, whose group built a castle from sugar
cubes, wrote: "I remember Nick proudly announcing to the other members of
the group, 'I need 28 cubes for this wall because it's 11 times 2 and you
have to add the 6 bumps!" Brad Uy's group studied volume and density,
grappled with objections that "air does not have volume," and built a model
boat christened "The Big Kahuna." Brendan Kinnell's group discovered
tessellations and, like M. C. Escher, created their own. Heather Stephens's
group studied colors and produced beautiful tonal artwork. Eric Bellucci's
group created and performed percussion rhythms. Hoyoon Nam's group
constructed their own instruments, wrote and recorded a new song, "The
Millennium Kids," and sent it in to the local radio station. Sarah
Schiavetti's group created a whole new "Greek" myth explaining the phases
of the moon, with new words invented from Greek roots, such as the
"aristoastros" (best stars). Elizabeth Healy's group wrote a play and
performed it. They also wrote poetry in the form of haikus and cinquains.
Here are a haiku and two cinquains they wrote:

**Butterflies**

Different colors,

Beautiful fluttering wings,

It lands on my nose,

Color.

--Katy Markland

** Willows**

Drooping over rivers

Touching pebbles and catfish

And tickling them.

--Chloe Dircks

**Whales**

Gracefully swimming

Diving in the salty sea

Splashing all around.

--Lian Larkin

As Chloe said, "It really calms me down to put my ideas into a poem." The groups presented stories and contests at the local Water Street Books and town library, and appeared with the contest winners on the live, call-in Math Chat TV show. They also submitted material to radio and magazines.

**OLD CHALLENGE.** Just last summer (see June 17 Math Chat) Thomas Hales of the
University of Michigan proved that regular hexagons provide the
least-perimeter (least-length) way to divide the plane into equal areas.
What is the shortest network of curves you can find dividing the surface of
the sphere into say four equal areas? five equal areas? other numbers of
equal areas?

**ANSWER.** For four equal areas, Joseph DeVincentis suggests a network based
on the regular tetrahedron, with four congruent spherical triangles meeting
in threes. For more general numbers of areas, he suggests networks based on
the various regular solids: not just the 4-faced tetrahedron, but also the
6-faced cube, and the 8-faced octahedron, not to mention the 12-faced
dodecahedron and the 20-faced icosahedron. He correctly notes that the
octahedron is not actually optimal, could be improved, as can any example
in which the curves meet in fours or more rather than in threes. (Similarly
the icosahedron could be improved.) The other three are conjectured to be
optimal for 4, 6, and 12 regions. Currently Christopher Chris Cox

For 5 regions, DeVincentis suggests a network based on the triangular prism. (Prisms and antiprisms might help for larger numbers of regions too.)

**NEW CHALLENGE.** Was there any validity to the claim that the full moon of
December 22, 1999 was the brightest that we shall see for millions of years?

Send answers, comments, and new questions by email to
Frank.Morgan@williams.edu, to be eligible for *Flatland* and other book
awards. Winning answers will appear in the next Math Chat. Math Chat
appears on the first and third Thursdays of each month. Prof. Morgan's
homepage is at www.williams.edu/Mathematics/fmorgan.

MATH CHAT TV, the live, call-in show with questions and prizes, will air on the web Wednesdays, February 9 and 16, 7-7:30 pm Eastern time, at raserver.williams.edu/live/MathChat.rm

THE MATH CHAT BOOK, including a $1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).

Copyright 2000, Frank Morgan.